#8926349

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(3,9)(8,15)(10,13)(11,16)(17,18,23,28)(19,26,21,24)(20,22,27,25)(29,32,31,30)(34,38)(36,43)(39,44)(42,47);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(17,19,23,21)(18,24,28,26)(20,31,27,29)(22,32,25,30)(35,41)(39,44)(42,47)(46,48);(2,7)(5,12)(8,15)(11,16)(17,20,23,27)(18,25,28,22)(19,29,21,31)(24,32,26,30)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,21,23,19)(18,26,28,24)(20,31,27,29)(22,32,25,30);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,22,23,25)(18,27,28,20)(19,30,21,32)(24,31,26,29)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833545);